Analytical Structure and Stability Analysis of a Fuzzy Two-Term Controller with Multi-Fuzzy Sets
Arpita Ghosh, T. K. Das, S. K. Mandal National Metallurgical Laboratory (CSIR)
Jamshedpur - 831007, India
B. M. Mohan
Indian Institute of Technology Kharagpur – 721302, India
Abstract
This paper reveals the analytical structure of a fuzzy two- term (PI/PD) controller which employs N1 (≥3) number of symmetric fuzzy sets for the input variable ‘displacement’, N2 (≥3) number of symmetric fuzzy sets for the input variable ‘velocity’ and N1+N2-1 number of symmetric fuzzy sets for the output variable ‘controller output’. The analytical structures are derived via triangular membership functions for fuzzification of the inputs and output variables, linear control rules, minimum triangular norm, algebraic sum triangular co-norm, different inference methods and center of sums (COS) defuzzification method and properties of such structures are investigated. Using the well-known small-gain theorem, bounded-input bounded-output (BIBO) stability analysis of feedback systems involving fuzzy PD controller as a subsystem is presented. Finally, a numerical example along with its simulation results is included to validate the effectiveness of the fuzzy two-term controller.
1. Introduction
As it appears from the literature, Mizumoto [2] has investigated fuzzy control problem by considering multi- fuzzy sets and different fuzzy reasoning methods. It has been shown [4] that a fuzzy controller can be designed in such a manner that it is at least as good as the conventional PID controller which allows the plant under control to follow a specified behaviour. In [5], a fuzzy PI controller with triangular fuzzy numbers, Zadeh logic to evaluate linear control rules, and center of gravity defuzzifier has been considered. A closed form expression of the defuzzified output has been derived and shown it to be a nonlinear controller. Also, the nonlinearities of the fuzzy controller have been analyzed. Using an arbitrary number of inputs and an arbitrary number of triangular fuzzy sets to fuzzify every input variable, and probability AND operator and Lukasiewicz OR operator to formulate the fuzzy control rules, it has been shown [6] that the defuzzfied output of the controller becomes a linear parametric function of the controller inputs.
A multi-region fuzzy logic controller for nonlinear processes was proposed [7]. Based on apriori knowledge, the process to be controlled was divided into fuzzy regions such as high-gain, low-gain, large-time-constant, and small-
time-constant. Then a fuzzy controller was designed based on the regional information. A set of linguistic rules was formulated [8] in terms of their governing equations and asymptotic control laws were proposed for a small input error. Also, the effect of design factors like inference operators and number of partitions were studied. For a single input and single output (linear or nonlinear) system, it was shown [9] that one could construct a fuzzy logic controller equivalent to a given PI controller, and that a fuzzy logic controller designed with prescribed fuzzy logic operations was essentially a PI controller.
The input-output parametric relationship of a class of crisp-type fuzzy logic controllers using various t-norm sum- gravity inference methods has been studied [11]. Using four most important t-norms, the matching level of each control rule has been calculated, and the explicit mathematical forms of reasoning surfaces have been obtained. The reasoning surfaces of these crisp-type fuzzy controllers have been proved to be composed of a 2-D multi-level relay and a local position-dependent nonlinear compensator with output pattern being influenced by the t-norm selected.
Input-output structures of fuzzy controllers with nonlinear input fuzzy sets, singleton output fuzzy sets, product AND operator, Mamdani type of fuzzy rules, and centroid defuzzifier were studied [13]. In addition, conditions for these structures to be equivalent to nonlinear PI/PD controllers with variable gains were provided. Recently, attempts have been made [14] to develop analytical structures for fuzzy PI/PD controllers with arbitrary trapezoidal input fuzzy sets having the property: the sum of two neighboring membership functions is not equal to unity.
The analytical structures in [5, 6, 11, 13, 14] have been derived using either maximum or bounded sum triangular conorm.
It is evident from the literature survey that application of algebraic sum triangular co-norm for fuzzy two-term control is yet to be explored. Therefore, the main objectives of this paper are (i) to derive analytical structures of fuzzy two-term (PI/PD) controllers by employing minimum triangular norm, algebraic sum triangular co-norm, N1
number of symmetric triangular membership functions on the input variable „displacement‟, N2 number of symmetric triangular membership functions on the input variable
„velocity‟, N1+N2-1 number of symmetric triangular membership functions on output, linear control rules,
Mamdani minimum / Larsen product / drastic product inference method, and COS method of defuzzification, (ii) to investigate the properties of the controllers derived in (i), and (iii) to establish BIBO stability conditions for the feedback system containing the fuzzy PD controller as the subsystem. This paper is organized as follows: Section 2 deals with scaling factors, fuzzification and defuzzification modules, control rule base, and inference engine; Section 3 presents analytical structures of fuzzy PI controllers with symmetric triangular fuzzy sets; properties of fuzzy PI controllers with symmetric triangular fuzzy sets are discussed in Section 4; Section 5 shows that the analytical structures and their properties presented for fuzzy PI controllers also hold good for fuzzy PD controllers. BIBO stability analysis of fuzzy PD control systems is given in Section 6; Section 7 includes the results of simulation studies done on a linear second-order time delay system.
The last section concludes the paper.
2. Fuzzy PI /PD controller
The principal structure of a fuzzy PI / PD controller is shown in Figure 1 which consists of the components such as scaling factors, fuzzification and defuzzification modules, rule base and inference engine. The incremental control signal (velocity algorithm) [3] generated by discrete-time PI controller is given by
∆𝑢 𝑘𝑇 = 𝑢 𝑘𝑇 − 𝑢[ 𝑘 − 1 𝑇
= 𝐾𝑃𝑑 𝑣 𝑘𝑇 + 𝐾𝐼𝑑 𝑑(𝑘𝑇 (1) while the control effort produced by discrete-time PD controller is given by
𝑢(𝑘𝑇) = 𝐾𝑃𝑑 𝑑(𝑘𝑇) + 𝐾𝐷𝑑𝑣(𝑘𝑇) (2) where 𝐾𝑃𝑑, 𝐾𝐼𝑑, and 𝐾𝐷𝑑 are respectively the proportional, integral and derivative constants of discrete-time PI and PD controllers,
𝑑 𝑘𝑇 = 𝑒 𝑘𝑇 , the displacement (3) and 𝑣 𝑘𝑇 = 𝑑 𝑘𝑇 −𝑑 𝑘− 1 𝑇
𝑇 , the velocity (4) 𝑒(𝑘𝑇) is the error signal, 𝑇 is the sampling period. Based on (1), (2) and (4), the principal structures of fuzzy PI controller and fuzzy PD controller are shown in Figure 1, in which 𝑁𝑑, 𝑁𝑣, 𝑁∆𝑢−1 and 𝑁𝑢−1 represent scaling factors of the fuzzy controllers, and 𝑑𝑁(𝑘𝑇), 𝑣𝑁(𝑘𝑇), ∆𝑢𝑁(𝑘𝑇) and 𝑢𝑁(𝑘𝑇) represent normalized versions of 𝑑(𝑘𝑇), 𝑣(𝑘𝑇),
∆𝑢(𝑘𝑇) and 𝑢(𝑘𝑇), respectively.
2.1. Scaling factors
The use of normalized universes of discourse requires a scale transformation which maps the physical values of the process state variables (in the present situation 𝑑(𝑘𝑇) and
𝑣(𝑘𝑇)) into a normalized domain. This is called input normalization. Furthermore, output denormalization maps the normalized value of the output variable (here ∆𝑢𝑁(𝑘𝑇) or 𝑢𝑁(𝑘𝑇)) into its physical domain. The scaling factors which describe the inputs normalization (𝑁𝑑 and 𝑁𝑣) and output denormalization (𝑁∆𝑢−1 or 𝑁𝑢−1) play a role similar to that of the gain co-efficients (𝐾𝑃𝑑, 𝐾𝐼𝑑, and 𝐾𝐷𝑑) in conventional controllers.
Figure 1. Block diagram fuzzy PI/PD control system 2.2. Fuzzification module
It converts instantaneous value of a process state variable into a linguistic value with the help of represented fuzzy set. The parametric functional description of the triangular shaped membership function is the most economic one and hence it is considered here. Let the number of fuzzy sets, 𝑁 on scaled input variables “displacement 𝑑𝑁(𝑘𝑇)” and
“velocity 𝑣𝑁(𝑘𝑇)” be the same, and the membership functions be identical. Assume that there are J number of fuzzy sets on negative displacement (velocity), one fuzzy set for zero displacement (velocity), and 𝐽number of fuzzy sets on positive displacement (velocity). Therefore, there is a total of
𝑁 = 2𝐽 + 1 ≥ 3 (5) number of fuzzy sets on each input variable as shown below:
{𝑋−𝐽, 𝑋− 𝐽 −1 , ⋯ , 𝑋−1, 𝑋0, 𝑋1, ⋯ , 𝑋𝑝, ⋯ , 𝑋(𝐽 −1), 𝑋𝐽} (6) where 𝑋 is 𝐷 (for displacement) or 𝑉(for velocity). The membership functions corresponding to members in (6) are considered as
{𝜇−𝐽 𝑥𝑁 , 𝜇− 𝐽 −1 𝑥𝑁 , ⋯ , 𝜇−1 𝑥𝑁 , 𝜇0 𝑥𝑁 ,
𝜇1(𝑥𝑁), ⋯ , 𝜇𝑝(𝑥𝑁), ⋯ , 𝜇 𝐽 −1 𝑥𝑁 , 𝜇𝐽(𝑥𝑁)} (7) where 𝑥𝑁 is 𝑑𝑁 (with p = i) or 𝑣𝑁 (with p = j). Let the central value of membership function 𝜇𝑝 𝑥𝑁 be 𝜆𝑝, and define 𝜆−𝐽= −𝑙¸ 𝜆0= 0 and 𝜆𝐽 = 𝑙. Also, let the space S between central values of two adjacent members be equal.
Then S is given by S = 𝑙
𝐽 (8) Consequently, the central value 𝜆𝑝 becomes 𝜆𝑝= p . S.
Note that the base of each member is 2S. The membership function 𝜇𝑝 𝑥𝑁 is defined as follows:
Figure 2. Membership functions for input 𝒙𝑵
Figure 3. Output membership functions For 𝑝 = − 𝐽 − 1 , − 𝐽 − 2 , ⋯ , 𝐽 − 2 , and 𝐽 − 1
𝜇𝑖 𝑥𝑁 =
0 , 𝑥𝑁≤ (𝑝 − 1)𝑆
1
𝑆{𝑥𝑁− (𝑝 − 1)𝑆} , (𝑝 − 1)𝑆 ≤ 𝑥𝑁≤ 𝑝𝑆 −1𝑆 𝑥𝑁− 𝑝 + 1 𝑆 , 𝑝𝑆 ≤ 𝑥𝑁≤ (𝑝 + 1)𝑆
0 , (𝑝 + 1)𝑆 ≤ 𝑥𝑁
(9)
For 𝑝 = −𝐽
𝜇−𝐽 𝑥𝑁 =
0 , 𝑥𝑁≤ −𝐿 1, −𝐿 ≤ 𝑥𝑁≤ −𝐽𝑆
−1
𝑆 𝑥𝑁− −𝐽 + 1 𝑆 , −𝐽𝑆 ≤ 𝑥𝑁≤ −𝐽 + 1 𝑆 0 , (−𝐽 + 1)𝑆 ≤ 𝑥𝑁
(10)
For 𝑝 = 𝐽
𝜇𝐽 𝑥𝑁 =
0 , 𝑥𝑁≤ (𝐽 − 1)𝑆 1𝑆{𝑥𝑁− (𝐽 − 1)𝑆} , (𝐽 − 1)𝑆 ≤ 𝑥𝑁≤ 𝐽𝑆
1, 𝐽𝑆 ≤ 𝑥𝑁≤ 𝐿 0 , 𝐿 ≤ 𝑥𝑁
(11)
Notice that
𝜇𝑝 𝑥𝑁 + 𝜇𝑝+1 𝑥𝑁 = 1, 𝑥𝑁 𝜖 [−𝐿, 𝐿] (12)
𝑖 + 𝑗 = 𝑚 (13) Figure 2 shows membership function 𝜇𝑝 𝑥𝑁 corresponding to the input fuzzy set 𝑋𝑝 in (6). Let 𝑥𝑁 be the crisp input. Then from Figure 2 the fuzzified version of 𝑥𝑁 is its degree of membership 𝜇𝑝 𝑥𝑁 . Assume that there are 2N − 1 (i.e. 4J+1) number of fuzzy sets on the normalized output variable ∆𝑢𝑁 𝑘𝑇 for PI and 𝑢𝑁 𝑘𝑇 for PD). Among these, 2J members are on negative output, one member for zero output, and 2J members are on
positive output. The membership functions for normalized output, shown in Figure 3, are denoted by
𝑂−𝐽, 𝑂−(𝐽 −1), ⋯ , 𝑂−1, 𝑂0, 𝑂1, ⋯ , 𝑂𝑚, ⋯ , 𝑂(𝐽 −1), 𝑂𝐽 (14) Let the central value of member 𝑂𝑚 be 𝛾𝑚 and define 𝛾−2𝐽
=-H, 𝛾0= 0, and 𝛾2𝐽= H. Further, let the space W between the central values of two adjacent members be equal. Then
W = 𝐻
2𝐽 = 𝑁−1𝐻 (15) and 𝛾𝑚 = 𝑚. 𝑊 =𝑁−1𝑚𝐻 (16) 2.3. Control rule base
The following linear control rules are considered for fuzzy PI controllers:
R1) If 𝑑𝑁 is 𝐷𝑖+1 AND 𝑣𝑁 is 𝑉𝑗 then ∆𝑢𝑁 is 𝑂𝑚 +1. R2) If 𝑑𝑁 is 𝐷𝑖 AND 𝑣𝑁 is 𝑉𝑗 then ∆𝑢𝑁 is 𝑂𝑚. R3) If 𝑑𝑁 is 𝐷𝑖 AND 𝑣𝑁 is 𝑉𝑗 +1 then ∆𝑢𝑁 is 𝑂𝑚 +1. R4) If 𝑑𝑁 is 𝐷𝑖+1 AND 𝑣𝑁 is 𝑉𝑗 +1 then ∆𝑢𝑁 is 𝑂𝑚 +2. The above control rules also hold good for fuzzy PD controller if ∆𝑢𝑁 is replaced by 𝑢𝑁. The AND operation considered in the rule base is minimum triangular norm, which is defined as:
𝜇 𝑚𝑖𝑛(𝑑𝑁, 𝑣𝑁) = min(𝜇𝑎 𝑑𝑁 , 𝜇𝑏(𝑣𝑁)) (17)
where 𝑎 𝜖 {𝐷𝑖, 𝐷𝑖+1} and 𝑏 𝜖 {𝑉𝑗, 𝑉𝑗 +1} are the 𝑎𝑡ℎ and 𝑏𝑡ℎ fuzzy sets on 𝑑𝑁 and 𝑣𝑁 respectively.
Table 1. Inference methods
Inference Method Definition Area of inferred output
fuzzy set 𝐀(𝛍 )
Mamdani Minimum, RMM min(𝜇 , 𝜇(𝑜𝑢𝑡𝑝𝑢𝑡)) 𝜇 W(2-𝜇 )
Larsen Product, RLP 𝜇 . 𝜇(𝑜𝑢𝑡𝑝𝑢𝑡) 𝜇 W
Drastic Product, RDP Case 1: 𝜇 , if 𝜇 𝑜𝑢𝑡𝑝𝑢𝑡 = 1 Case 2: 𝜇 𝑜𝑢𝑡𝑝𝑢𝑡 if 𝜇 = 1 Case 3: 0 if 𝜇 , 𝜇 𝑜𝑢𝑡𝑝𝑢𝑡 >0
− 𝑊 0
Figure 4. Illustration of inference methods
Figure 5. Possible input combinations of 𝒅𝑵(𝒌𝑻) and 𝒗𝑵(𝒌𝑻) (a) in the region: 𝒊𝑺 ≤ 𝒅𝑵 𝒌𝑻 ≤ (𝒊 + 𝟏)𝑺 and 𝒋𝑺 ≤ 𝒗𝑵 𝒌𝑻 ≤ (𝒋 + 𝟏)𝑺 (b) outside the region:
𝒊𝑺 ≤ 𝒅𝑵 𝒌𝑻 ≤ (𝒊 + 𝟏)𝑺 and 𝒋𝑺 ≤ 𝒗𝑵 𝒌𝑻 ≤ (𝒋 + 𝟏)𝑺
2.4. Inference engine
The degree of match between the crisp input and the fuzzy sets describing the meaning of the rule-antecedent is computed for each rule using minimum triangular norm defined in (17). Then the degree of match is used to
determine the inferred output fuzzy set via any of the fuzzy inference methods defined in Table 1, and graphically illustrated in Figure 4.
There are four possible input combinations (ICs), see Figure 5(a), of the normalized inputs 𝑑𝑁(𝑘𝑇) and 𝑣𝑁(𝑘𝑇) in the region defined by 𝑖𝑆 ≤ 𝑑𝑁 𝑘𝑇 ≤ (𝑖 + 1)𝑆 and 𝑗𝑆 ≤ 𝑣𝑁 𝑘𝑇 ≤ (𝑗 + 1)𝑆. Similarly there are eight possible input combinations of the normalized inputs in the region shown in Figure 5(b).
The control rules in Section 2.3 are used to evaluate appropriate control law in each IC region. The outcomes of the control rules for all the IC regions with minimum triangular norm are listed in Table 2. It may be noticed from the control rule base that the control rules R1 and R3 fire the same output fuzzy set Om+1, and produce two membership functions 𝜇 1(𝑜𝑢𝑡𝑝𝑢𝑡) and 𝜇 3(𝑜𝑢𝑡𝑝𝑢𝑡). To take care of the OR operation between rule 1 and rule 3, algebraic sum triangular co-norm is used, which is defined as follows:
𝜇1/3= 𝜇 1 + 𝜇 3 -𝜇 1 𝜇 3 (18)
2.5. Defuzzification
The well-known COS method of defuzzification is used to obtain the crisp controller output which is defined [10] as
∆𝑢𝑁 𝑘𝑇 =𝐴 𝜇 1 ℎ1+𝐴 𝜇 2 ℎ2+𝐴 𝜇 3 ℎ3+𝐴 𝜇 4 ℎ4
𝐴 𝜇 1 +𝐴 𝜇 2 +𝐴 𝜇 3 +𝐴 𝜇 4 (19) where ℎ𝑖, 𝑖=1, 2, 3 and 4 is the centroid of inferred output fuzzy set corresponding to the 𝑖th rule. ∆𝑢𝑁 𝑘𝑇 is replaced by 𝑢𝑁 𝑘𝑇 for fuzzy PD controller. From the control rule base, it can be seen that the output fuzzy set, Om+1 is fired two times (see rules R1 and R3). In such a situation, using
algebraic sum triangular co-norm defined in (18), (19) can be written as
∆𝑢𝑁 𝑘𝑇 =𝐴 𝜇 2 ℎ2+𝐴 𝜇 1/3 ℎ1/3+𝐴 𝜇 4 ℎ4
𝐴 𝜇 2 +𝐴 𝜇 1/3 +𝐴 𝜇 4 (20) where 𝜇 1/3 is the outcome of triangular co-norm and h is the centroid of the inferred output fuzzy set shown (with hatching) in Figure 4.
Since, the output fuzzy sets are symmetrical about their central values 𝛾m‟s, the global centroid can be calculated from the local centroids. The inferred area of respective output fuzzy set weighs the importance of each local centroid in the global centroid. Considering each of the local centroids shown in Figure 6, (20) can be written as
∆𝑢 𝑘𝑇 = 𝑁1
∆𝑢
𝐴 𝜇 2 𝑚𝑊 +𝐴 𝜇 1/3 𝑚 +1 𝑊+𝐴 𝜇 4 𝑚 +2 𝑊 𝐴 𝜇 2 +𝐴 𝜇 1/3 +𝐴 𝜇 4 = 𝑁1
∆𝑢
𝐴 𝜇 2 𝑖+𝑗 𝑊+𝐴 𝜇 1/3 𝑖+𝑗 +1 𝑊+𝐴 𝜇 4 (𝑖+𝑗 +2)𝑊 𝐴 𝜇 2 +𝐴 𝜇 1/3 +𝐴 𝜇 4 (21)
Table 2. Outcomes of rules in IC regions
Figure 6. Center of sums defuzzification method (for RLP inference method)
3. Analytical structures of the fuzzy PI controllers with symmetric triangular fuzzy sets
Here analytical structures of three classes of fuzzy PI controllers with multiple symmetric fuzzy sets are presented. The expression in (21) can be written as
∆𝑢 𝑘𝑇 =(𝑖+𝑗 +1)𝑊𝑁
∆𝑢 + 𝑁𝑊
∆𝑢
𝐴 𝜇 4 −𝐴 𝜇 2
𝐴 𝜇 2 +𝐴 𝜇 1/3 +𝐴 𝜇 4 =∆𝑢𝑔+ ∆𝑢𝑙 (22) where ∆𝑢𝑔 =(𝑖+𝑗 +1)𝑊𝑁
∆𝑢 (23) and ∆𝑢𝑙 = 𝑁𝑊
∆𝑢
𝐴 𝜇 4 −𝐴 𝜇 2
𝐴 𝜇 2 +𝐴 𝜇 1/3 +𝐴 𝜇 4 (24) For simplicity we define
𝑧1 𝑘 = 𝑑𝑁 𝑘 − (𝑖 + 0.5)𝑆 (25) 𝑧2 𝑘 = 𝑣𝑁 𝑘 − 𝑗 + 0.5 𝑆 (26) and show 𝑘𝑇 as 𝑘 in the sequel. The analytical structures of different classes of fuzzy PI controllers follow now.
3.1. Analytical structures in the regions IC1 - IC4
Class I: Triangular norm: minimum, triangular co-norm:
algebraic sum, inference method: Mamdani minimum.
∆𝑢𝑙(𝑘) = 𝑁𝑊
∆𝑢
𝑆+ 𝑧1 𝑘 −𝑧2 𝑘 (𝑧1 𝑘 +𝑧2 𝑘 ) 3𝑆2−2𝑆 𝑧 𝑘 −2 𝑧12 𝑘 +𝑧22 𝑘 −𝑆2𝑊(0.5625 𝑆4
−0.75𝑆3 𝑧1 𝑘 +𝑧2 𝑘 −0.75𝑆2(𝑧12 𝑘 +𝑧22(𝑘)) +𝑆2𝑧1 𝑘 𝑧2 𝑘 +𝑆𝑧1 𝑘 𝑧2 𝑘 𝑧1 𝑘 +𝑧2 𝑘
+𝑧12 𝑘 𝑧22(𝑘))
(27)
where 𝑧(𝑘) is defined in Table 3.
Class II: Triangular norm: minimum, triangular co-norm:
algebraic sum, inference method: Larsen product.
∆𝑢𝑙 𝑘 = 𝑁𝑊
∆𝑢
𝑧1 𝑘 +𝑧2 𝑘
2𝑆−2 𝑧 𝑘 −𝑊𝑆(0.25𝑆2−0.5𝑆 𝑧1 𝑘 +𝑧2 𝑘 +𝑧1 𝑘 𝑧2 𝑘
(28)
where 𝑧(𝑘) is defined in Table 3.
Class III: Triangular norm: minimum, triangular co-norm:
algebraic sum, inference method: drastic product.
∆𝑢𝑙(𝑘) = 𝑁𝑊
∆𝑢
𝑧1 𝑘 +𝑧2 𝑘
2𝑆−2 𝑧 𝑘 −1𝑆(0.25𝑆2−0.5𝑆 𝑧1 𝑘 +𝑧2 𝑘 +𝑧1 𝑘 𝑧2 𝑘 )
(29) where 𝑧(𝑘) is defined in Table 3.
Table 3. Attributes of 𝒛 𝒌
𝒛(𝒌) 𝑰𝑪𝒔
𝑧1(𝑘) 𝑧2(𝑘)
𝐼𝐶1 , 𝐼𝐶3 𝐼𝐶2 , 𝐼𝐶4
For all classes, ∆𝑢𝑔 in regions IC1- IC4 is given by (23).
3.2. Analytical structures in the regions IC5 – IC8
Class I, class II and class III:
∆𝑢𝑔 = 𝐽 +𝑗 𝑊 𝑁
∆𝑢 in the region IC5 (30)
∆𝑢𝑔 = 𝑖−𝐽 +1 𝑊 𝑁
∆𝑢 in the region IC6 (31)
∆𝑢𝑔= −𝐽+𝑗 +1 𝑊
𝑁∆𝑢 in the region IC7 (32)
∆𝑢𝑔= 𝑖+𝐽 𝑊 𝑁
∆𝑢 in the region IC8 (33) Class I:
∆𝑢𝑙(𝑘) = 2𝑁𝑊
∆𝑢
𝑆𝑧2 𝑘
0.75𝑆2−𝑧22 𝑘 + 1 in the region IC5 (34)
∆𝑢𝑙(𝑘) = 2𝑁𝑊
∆𝑢
𝑆𝑧1 𝑘
0.75𝑆2−𝑧12 𝑘 − 1 in the region IC6 (35)
∆𝑢𝑙(𝑘) = 2𝑁𝑊
∆𝑢
𝑆𝑧2 𝑘
0.75𝑆2−𝑧22 𝑘 − 1 in the region IC7 (36) ∆𝑢𝑙(𝑘) = 2𝑁𝑊
∆𝑢
𝑆𝑧1 𝑘
0.75𝑆2−𝑧12 𝑘 + 1 in the region IC8 (37) Class II and class III:
∆𝑢𝑙(𝑘) = 2𝑁𝑊
∆𝑢 2𝑧2 𝑘
𝑆 + 1 in the region IC5 (38) ∆𝑢𝑙(𝑘) = 2𝑁𝑊
∆𝑢 2𝑧1 𝑘
𝑆 − 1 in the region IC6 (39) ∆𝑢𝑙(𝑘) = 2𝑁𝑊
∆𝑢 2𝑧2 𝑘
𝑆 − 1 in the region IC7 (40) ∆𝑢𝑙(𝑘) = 2𝑁𝑊
∆𝑢 2𝑧1 𝑘
𝑆 + 1 in the region IC8 (41) 3.3. Analytical structures in the regions IC9 - IC12
Class I, class II and class III:
∆𝑢(𝑘) = 0 in the region IC9 and IC11 (42)
∆𝑢(𝑘) = 𝑁−𝐻
∆𝑢 in the region IC10 (43)
∆𝑢(𝑘) = 𝑁𝐻
∆𝑢 in the region IC12 (44)
4. Properties of fuzzy PI controllers with symmetric triangular fuzzy sets
In (22) ∆𝑢𝑔 represents the global two-dimensional multi- level relay and ∆𝑢𝑙(𝑘) represents the local nonlinear PI controller. In order to examine the roles of global multilevel relay and local nonlinear PI controller in total control action, and the degree of nonlinearity of fuzzy controller as N changes, we define a constant 𝜂 as
𝜂 = ∆𝑢 ∆𝑢𝑙𝑚𝑎𝑥
𝑔𝑚𝑎𝑥+ ∆𝑢𝑙𝑚𝑎𝑥 (45) It can be seen from (45) that (i) as 𝑁 ≥ 3, for 𝑁 = 3 the value of η is 50% which implies that the global multilevel relay and the local nonlinear PI controller play equal role in total control action, and (ii) as N becomes larger, 𝜂 approaches a smaller value which makes the resolution of
global multi-level relay output finer and the fuzzy controller less nonlinear. Moreover, the global multi-level relay has a stronger role than the local nonlinear PI controller, in the total control action.
In view of the above analytical structures and their properties we have
Theorem 1: The structure of the fuzzy PI controller with linear control rules is the sum of a global two- dimensional multi-level relay and a local nonlinear PI controller.
5. Analytical structures of fuzzy PD controllers
Since it is clear from Figure 1 that 𝑑(𝑘𝑇) and 𝑣(𝑘𝑇) are always the inputs to the two-term (PI or PD) controller and
∆𝑢(𝑘𝑇) is the incremental output of the PI controller (velocity algorithm) while 𝑢(𝑘𝑇) is the output of the PD controller, the analytical structures for fuzzy PI controllers in Section 3 are equally applicable to even fuzzy PD controllers provided we replace ∆𝑢(𝑘𝑇) by 𝑢(𝑘𝑇), 𝑁∆𝑢−1 by 𝑁𝑢−1 and set 𝑢[(𝑘 − 1)𝑇] equal to zero. Also, the properties of different classes of fuzzy PI controllers in Section 4 hold good for fuzzy PD controllers.
6. BIBO stability analysis of feedback systems that contain fuzzy PD controllers
We make use of the well-known small-gain theorem in [1]
to derive sufficient conditions for BIBO stability of feedback systems which contain one of the fuzzy PD controllers in Section 3 as a subsystem. The sufficient condition for fuzzy PD control system to be BIBO stable can be stated as follows.
Figure 7. Block diagram of a typical fuzzy PD control system
Theorem 2: The system of Figure 7 is BIBO stable if (i) the given nonlinear process 𝑁2 has a bounded norm i.e., 𝑁2 < ∞ where . denotes the norm of nonlinear mapping, and (ii) the parameters 𝑆, 𝑊, 𝑇, 𝑁𝑑, 𝑁𝑣 and 𝑁𝑢 of fuzzy PD controller satisfy the inequality
𝑊× 𝑇×𝑁𝑑+𝑁𝑣
𝑆×𝑇×𝑁𝑢 𝑁2 < 1 (46) This theorem is applicable to all the three classes of control systems.
7. Validation of fuzzy PD controller
Comparison of the performances of linear PD controller and the fuzzy PD controller with symmetric input and output fuzzy sets is done here by considering the following example
A linear second-order time-delay system 𝐺𝑝 𝑠 = 𝑒−3𝑠
(100𝑠 + 1)2 (47)
with a unit setpoint. In (47), 𝐺𝑝 𝑠 represent the transfer function of the plant to be controlled. To implement the mathematical model of fuzzy PD controller, the design parameters are to be appropriately chosen. For this, the design procedure in [12] is followed here to design the fuzzy PD controller. For the above process, the values of sampling period T=0.1sec, proportional gain 𝐾𝑃𝑑= 6.0, derivative gain 𝐾𝐷𝑑 = 2.5, maximum absolute displacement (error) 𝑑 𝑚𝑎𝑥 = 1 and maximum absolute velocity 𝑣 𝑚𝑎𝑥 = 0.0151. Since the control performance with the calculated design parameters is not found to be satisfactory, the parameters of the fuzzy controller are then fine tuned on trial and error basis to attain acceptable performance. The parameters 𝑁, 𝑁𝑑, 𝑁𝑣, 𝑁𝑢, 𝑙 and 𝐻 of the fuzzy controllers are listed in Table 4.
The unit step responses obtained with different classes of fuzzy PD controllers for the plant 𝐺𝑝 𝑠 are shown in Figure 8. These figures also show the corresponding responses with the conventional PD controllers. From the plots it is apparent that the fuzzy PD controllers outperform the conventional PD controllers for different values of N. The basic objective of this simulation study is to demonstrate the influence of the parameter 𝑁 on the performance of the fuzzy controller. Therefore, for different classes of fuzzy PD controllers other functional
parameters are fixed and only 𝑁 is varied as shown in Table 4.
From the time-domain performance data in Table 4 it is observed that for class I and class III fuzzy PD controllers, by increasing the value of 𝑁 improved performance is obtained with drastic fall in the value of percent peak overshoot (%𝑀𝑝) and slight variation in the values of rise time (𝑡𝑟) and settling time (𝑡𝑠).
8. Conclusions
In this paper, analytical structures for different classes of fuzzy PI/PD controllers with multiple fuzzy sets have been considered. Using 𝑁1 number of symmetric triangular membership functions on the input variable
„displacement‟, 𝑁2 number of symmetric triangular membership functions on the input variable „velocity‟, 𝑁1+𝑁2− 1 number of symmetric triangular fuzzy sets for the output variable, linear control rules, minimum triangular norm, algebraic sum triangular co-norm, three different inference methods (RMM, RLP, and RDP) and COS defuzzification, expressions for control laws of fuzzy PI/PD controllers have been derived. It has been shown that each resulting controller is equivalent to the sum of a global two- dimensional multilevel relay and a local nonlinear PI/PD controller.
Upon carefully investigating the properties of the fuzzy controllers, it has been found that all the three classes of fuzzy controllers exhibit desirable control properties.
Using the small-gain theorem, BIBO stability conditions are established considering any class of the controller as a fuzzy PD controller. The superiority of fuzzy PD controller over the linear PD controller has been demonstrated through a simulation study on a linear second-order time- delay system.
Table 4. Attributes and time-domain performance data of plants with conventional and fuzzy PD controllers
Plant Controller Class 𝑵𝒅 𝑵𝒗 𝑵𝒖 𝒍 𝑯 𝑵 𝑴𝒑
(%) 𝒕𝒓 (sec)
𝒕𝒔 (sec)
𝐺𝑝(𝑠)
Linear PD - - - 27.94 59.1365 -
Fuzzy PD I I I
2.4 2.4 2.4
34.0 34.0 34.0
0.022 0.022 0.022
2.4 2.4 2.4
4.0 4.0 4.0
3 5 7
3.1833 2.4623 2.2740
14.3 16.2 15.2
20.4 22.4 21.4 II
II II
2.56 2.56 2.56
26.0 26.0 26.0
0.0085 0.0085 0.0085
3.2 3.2 3.2
4.0 4.0 4.0
3 5 7
6.9945 2.3737 2.4792
10.0 10.0 10.0
15.0 15.1 14.9 III
III III
2.21 2.21 2.21
24.0 24.0 24.0
0.006 0.006 0.006
3.0 3.0 3.0
3.0 3.0 3.0
3 5 7
4.655 0.0020 0.0001
11.0 11.0 11.0
16.0 17.0 17.0
Figure 8. Unit step response of the closed loop system with plant 𝑮𝒑 𝒔 and (a) Class I (b) Class II(c) Class III fuzzy PD controller
Acknowledgments
The authors are highly grateful to the anonymous referees for their helpful suggestions and encouragement. The authors would like to thank the Director, National Metallurgical Laboratory (CSIR) for his kind permission to publish this paper.
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